Compositing: When to use premultiplied alpha?
When should one check the Convert Premul box on nodes that use alpha?


Only when you know that your image has unassociated (often called "straight") alpha. Or in other words, when your image's alpha is not associated (premultiplied).


  why is there a Premul slider on the Alpha Over node? When would you
  need it to be something other than 0 or 1?


That's an unfortunate feature that brings a lot of confusion among users. It can be used to mitigate alpha errors. If you did your compositing right and didn't mix up associated and unassociated alphas, you should never use it.
And even if you didn't, that feature is a hack that will only mitigate, not fix those errors. 
It's better to track down your alpha to spot any mistake and fix it.


  I understand that pre-multiplied alpha is sometimes good and sometimes bad, and that if used improperly it can result in dark
  borders where transparency tapers off. But that's about all I know.
  Could you please enlighten me?


It's never bad. It's actually the best way transparency can be expressed in an RGB file as it allows both occlusion and emission (that means that your foreground can both block and emit light).
You can do most of your compositing without moving away from associated alpha and expect good results.
That doesn't necessarily mean that unassociated alpha is never useful. It has its uses, but CG and VFX are probably better suited for associated alpha.
You just have to know when your image is associated or unassociated so the correct alpha-over operation is used and it produces the correct results.